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11
An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
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An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
Typical ambiguity: trying to have your cake and eat it too
 the proceedings of the conference Russell 2001
"... Would ye both eat your cake and have your cake? ..."
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Interuniversal Teichmüller Theory IV: Logvolume Computations and Settheoretic Foundations, preprint
"... The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature mod ..."
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The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature models of conventional scheme theory”, called Θ ±ellNFHodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θdata. This data includes an elliptic curve EF over a number field F, together with a prime number l ≥ 5. Consideration of various properties of the logthetalattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGPmonoids”. Here, we recall that “multiradial algorithms ” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ ±ell NFHodge theater related to a given Θ ±ell NFHodge theater by means of a nonring/schemetheoretic horizontal arrow of the logthetalattice. In the present paper, estimates arising from these multiradial algorithms for splitting
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Enriched stratified systems for the foundations of category
"... This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much ..."
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This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have
FOUNDATIONS OF UNLIMITED CATEGORY THEORY: WHAT REMAINS TO BE DONE
"... Abstract. Following a discussion of various forms of settheoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unli ..."
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Abstract. Following a discussion of various forms of settheoretical foundations of category theory and the controversial question of whether category theory does or can provide an autonomous foundation of mathematics, this article concentrates on the question whether there is a foundation for “unlimited ” or “naive ” category theory. The author proposed four criteria for such some years ago. The article describes how much had previously been accomplished on one approach to meeting those criteria, then takes care of one important obstacle that had been met in that approach, and finally explains what remains to be done if one is to have a fully satisfactory solution. From the very beginnings of the subject of category theory as introduced by Eilenberg & Mac Lane (1945) it was recognized that the notion of category lends itself naturally to
LOGVOLUME COMPUTATIONS AND SETTHEORETIC FOUNDATIONS
, 2012
"... The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature mo ..."
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The present paper forms the fourth and final paper in a series of papers concerning “interuniversal Teichmüller theory”. In the first three papers of the series, we introduced and studied the theory surrounding the logthetalattice, a highly noncommutative twodimensional diagram of “miniature models of conventional scheme theory”, called Θ ±ellNFHodge theaters, that were associated, in the first paper of the series, to certain data, called initial Θdata. This data includes an elliptic curve EF over a number field F, together with a prime number l ≥ 5. Consideration of various properties of the logthetalattice led naturally to the establishment, in the third paper of the series, of multiradial algorithms for constructing “splitting monoids of LGPmonoids”. Here, we recall that “multiradial algorithms ” are algorithms that make sense from the point of view of an “alien arithmetic holomorphic structure”, i.e., the ring/scheme structure of a Θ ±ellNFHodge theater related to a given Θ ±ellNFHodge theater by means of a nonring/schemetheoretic horizontal arrow of the logthetalattice. In the present paper, estimates arising from these multiradial algorithms for splitting
SET THEORY FOR CATEGORY THEORY
, 810
"... Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical co ..."
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Abstract. Questions of settheoretic size play an essential role in category theory, especially the distinction between sets and proper classes (or small sets and large sets). There are many different ways to formalize this, and which choice is made can have noticeable effects on what categorical constructions are permissible. In this expository paper we summarize and compare a number
Exact completions and small sheaves
, 2012
"... We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κary exact categories” are a reflective sub2category of “κary sites”, for any regular cardinal κ. A κary exact category is an exact category with disjoint and universal κsmall ..."
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We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κary exact categories” are a reflective sub2category of “κary sites”, for any regular cardinal κ. A κary exact category is an exact category with disjoint and universal κsmall coproducts, and a κary site is a site whose covering sieves are generated by κsmall families and which satisfies a solutionset condition for finite limits relative to κ. In the unary
Three Conceptual Problems That Bug Me
, 1996
"... Introduction I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought ..."
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Introduction I will talk here about three problems that have bothered me for a number of years, during which time I have experimented with a variety of solutions and encouraged others to work on them. I have raised each of them separately both in full and in passing in various contexts, but thought it would be worthwhile on this occasion to bring them to your attention side by side. In this talk I will explain the problems, together with some things that have been tried in the past and some new ideas for their solution. Types of conceptual problems. A conceptual problem is not one which is formulated in precise technical terms and which calls for a definite answer. For this reason, there are no clearcut criteria for their solution, but one can bring some criteria to bear. These will vary from case to case. There are three general types of conceptual problems in mathematics of which the ones that I will discuss today are examples. These are: 1 ffi<F